Short proof of a metrization theorem∗ S. Moll
L.M. S´anchez Ruiz
Abstract In this note we provide a new insight into trans-separable spaces, i.e. those which are separable by seminorms. This approach enables getting an easy proof of the fact that in a wide class of uniform spaces, containing (DF ) and (LM )-spaces, precompact subsets are metrizable.
1
Introduction
A uniform space X is trans-separable [3] if every uniform cover [10, p. 210] of X admits a countable subcover. This general notion had already been used by Garnir, de Wilde and Schmets [5] as separable seminormed space and had proven to be useful while studying concrete uniform spaces problems including locally convex spaces (lcs), and more recently has been used by Robertson [8] in order to study the metrizability of (pre)compact sets in uniform spaces. His goal was an attempt to clarify the first of the following two fundamental questions which arise in a natural way concerning compactness in any lcs E: Question 1. When are the compact sets in E metrizable? Question 2. When is E weakly angelic? Partial answers were known since Pfister [6] proved that every precompact set in a (DF )-space is metrizable by using the fact that a lcs E is trans-separable if and only if each equicontinuous subset of the dual space E ′ of E is σ (E ′ , E)-metrizable. Supported by projects MTM2005-01182 and DPI2005-08732-C02-01 of the Spanish Ministry of Education and Science, co-financed by the European Community (FEDER projects). Received by the editors May 2006. Communicated by Y. F´elix. 1991 Mathematics Subject Classification : 46E40. Key words and phrases : Trans-separable space, metrizable subset. ∗
Bull. Belg. Math. Soc. Simon Stevin 14 (2007), 317–320
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Pfister’s result was extended by Valdivia [9, p. 68 (28)] in dual metric spaces by using quasi-Souslin spaces. Later on Cascales and Orihuela [2] studied the aforementioned two problems in a large class of lcs, called G which will be conveniently recalled hereafter, and include (LF )- and (DF )-spaces. Their approach provided a general result [2, Theorem 1] stating that all precompact sets in a lcs in class G are metrizable and every space in class G is weakly angelic. They also showed that if E is a lcs in class G and K is a weakly compact set in E contained in a separable subspace of E, then K is metrizable [2, 1.14] by using K-analytic structures connected with ordered families of compact sets in uniform spaces. Cascales, K¸akol and Saxon [1, Theorem 3.1] provided some alternative proofs for (LM)-spaces and dual metric spaces. A different form of Cascales-Orihuela’s theorem was presented by Robertson [8] who proved that a uniform space is trans-separable whenever it is covered by an adequately ordered family of precompact sets, applying this result to show that precompact sets in a lcs E are metrizable if its topological dual E ′ , endowed with the topology Tpr of uniform convergence on precompact subsets of E, is covered by an ordered family of Tpr precompact subsets. The aim of this note is to apply Robertson’s result in order to provide a new and simple proof of the following result of Cascales and Orihuela [2] whose original proof was based on some results of K-analytic spaces, angelic spaces and lower semi-continuous maps. Theorem 1.n Let (X, U) obe a uniform space such that the uniformity U has got a base B = Nα : α ∈ NN verifying: (*) for every α, β ∈ NN , with α ≤ β (that is with α(n) ≤ β(n) for each n) it holds Nβ ⊂ Nα . Then the precompact subsets of (X, U) are metrizable in the induced uniformity. n
Let us recall that a lcs belongs to class G if there is a family A = Aα : α ∈ NN of subsets of E ′ , called its G-representation, such that:
o
1. A covers E ′ ; 2. the Aα satisfy condition (*) above; 3. sequences in each Aα are equicontinuous. Class G enjoys good stability properties and includes many important lcs classes such as dual metric spaces (hence (DF )-spaces), (LM)-spaces (hence (LF )-spaces), the space of distributions D ′ (Ω) and real analytic A (Ω)-spaces for open Ω ⊂ RN , [1, 2]. Condition (c) implies that each Aα is countably σ (E ′ , E)-relatively compact, hence Tpr -precompact. Finally note that trans-separability for a lcs E means exactly that E is separable with respect to every continuous seminorm on E, or equivalently, for every neighbourhood of the origin U in E there exists a countable subset N in E such that E = N + U, [6]. Pfister also noted the easy and useful following observation. Lemma 2. Let E be a lcs. Then the space E is trans-separable iff every equicontinuous set in E ′ is metrizable in the weak topology σ(E ′ , E).
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Proof of Theorem
buc Let Cpr (X) be the lcs of bounded real-valued uniformly continuous functions defined on X with the topology of uniform convergence on precompact subsets of X, N and for each α = (m, α1 , α2 , ..., αn , ...) ∈ N× NN let us denote by Aα the set formed by those f ∈ Cpbuc (X) such that
kf k∞ ≤ m, |f (s) − f (t)| ≤ 1/n for (s, t) ∈ (X × X) ∩ Nαn , n ∈ N.
N
buc Clearly Aα ⊂ Aβ whenever α ≤ β and Cpr (X) = {Aα : α ∈ N× NN }. Note that each Aα is pointwise compact and, by equicontinuity, the pointwise buc topology and the induced topology by Cpr (X) coincide on each Aα , therefore each buc Aα is a compact subset of Cpr (X). By the aforementioned Robertson’s result [8] buc it follows that Cpr (X) is trans-separable. Hence by Lemma 2 each equicontinuous ′ ′ buc buc buc subset of the dual Cpr (X) is σ Cpr (X) , Cpr (X) −metrizable. Finally let K be a precompact subset of X and let us consider
S
buc W := {f ∈ Cpr (X) : sup |f (x)| ≤ 1}. x∈K
buc Let ϕ be the evaluation map from X into Cpr (X)′ , which is an isomorphism onto ϕ (X) provided with the topology induced by the weak∗ -topology by the construction given in [4, Section 6.8]. From the equality ϕ (K) = W ◦ ∩ ϕ (X) , it follows that K is metrizable.
Remark 1. Cascales and Orihuela approach to Question 1 [2, Theorem 2] uses Theorem 1 applied to the topology τ ′ of uniform convergence on the sets Aα . It turns out that for quasi-barrelled spaces in class G the argument is even easier since then τ ′ is coarser than τ and, both topologies coinciding on any τ -precompact subset, the proven theorem may be applied.
References [1] Cascales, B., K¸akol, J. and Saxon, S.A., Weight of precompact sets and tightness. J. Math. Anal. Appl. 269 (2002), 500–518. [2] Cascales, B. and Orihuela, J., On Compactness in Locally Convex Spaces. Math. Z. 195 (1987), 365–381. [3] Drewnowski, L., Another note on Kalton’s theorem. Studia Math. 52 (1975), 233–237. [4] K¨othe, G., Topological Vector Spaces I. Springer Verlag, Berlin Heidelberg 1969. [5] Garnir, H.G., de Wilde, M. and Schmets, J., Analyse Fonctionelle I. Basel, Birkh¨auser Verlag 1968. [6] Pfister, H.H., Bemerkungen zum Satz u ¨ber die Separabilit¨at der Fr´echet-MontelR¨aume. Arch. Math. 27 (1976), 86–92.
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[7] Robertson, A.P. and Robertson, W., Topological vector spaces. Cambridge University Press 1964. [8] Robertson, N., The metrisability of precompact sets. Bull. Austral. Math. Soc. 43 (1991), 131–135. [9] Valdivia, M., Topics in Locally Convex Spaces. Math. Studies 67, NorthHolland Amsterdam New York Oxford 1982. [10] Wilansky, A., Topology for Analysis. Robert E. Krieger Publ. Co. 1983.
ETSIA-Departamento de Matematica Aplicada, Universidad Politcnica de Valencia, E-46022 Valencia, Spain email:
[email protected] ETSID-Departamento de Matematica Aplicada, Universidad Politcnica de Valencia, E-46022 Valencia, Spain email:
[email protected]
